An understanding of time variation of voltages and currents in electrical networks is especially important in control systems, which include such electrical networks for processing measurements and manipulating control elements in real-time. Real-time control systems having electrical networks are used in a wide variety of applications, including temperature controls for heating and cooling units; speed, altitude, pitch and yaw controls for aircraft and spacecraft; etc. There are many more applications, from the exotic to the mundane—from medical devices to kitchen appliances.
In order to understand the time varying voltages and currents for a control system, the system's electrical networks must be modelled and analysed. Such an electrical network includes interconnected circuits of analog electrical elements such as resistors, inductors, capacitors, diodes, etc. A number of electrical laws apply to such electrical networks, in general. These include Kirchoff's current law, Kirchoff's voltage law, Ohm's law, the Y-delta transform, Norton's theorem, and Thevenin's theorem. An application of these laws results in a set of linear simultaneous equations for currents and voltages.
If the network has elements such as inductors and capacitors whose voltage-current relationship is described by linear operators, such as differential, integral, or integro-differential operators, then designers often use the well-known Laplace transform method in order to get the network equations into the form of linear algebraic equations. That is, the designers convert integro-differential network equations into simultaneous linear algebraic equations in the Laplace plane, which are generally simpler to solve. If the network has elements whose current-voltage relationship is governed by non-linear operators (algebraic, differential, integral, etc.), then designers dealing with such systems often linearize the equations about the desired operating point in order to study the system's off-design behaviour around the operating point.
Once the network equations have been set out as linear algebraic equations, the well-known Gaussian elimination method is often used to obtain a solution to the equations. If it was not necessary to transform the network equations to the Laplace plane in order to get them in linear algebraic form, and if all the network elements have known values, then the solution generally yields numeric values for network voltages and currents in the time domain. If the network equations were transformed to the Laplace plane in order to put them in linear algebraic form, a solution in the Laplace plane can be determined, such as by Gaussian elimination with respect to the transformed equations, and then an inverse Laplace transform can be applied to the solution in the Laplace plane. This transforms the equations from the Laplace plane back to the original, time-based domain. The solution to the time-domain equations yields numerical values for the network currents and voltages in the time domain; provided, once again, that particular values are known for each network element.
If particular values are not known for each network element, the solutions to the linear algebraic network equations will also generally be in algebraic form. Further, if the Laplace transform method has been applied to obtain the network equations in linear algebraic form, then the coefficient matrix for the equations in the Laplace plane will be algebraic due to the presence of the Laplace variable, even if numerical values of all the network elements are known. Hence the solution to the network equations in the Laplace plane will also be algebraic, even if numerical values of all the network elements are known.
Regardless of whether an electrical network is modelled by linear network equations or a linearized version of non-linear network equations, the branch voltages and currents of the network must satisfy respective sets of equations for three constraints: Kirchoff's current law; Kirchoff's voltage law; and branch voltage-current relationships. For a network having Nv nodes and Nb branches, the set for Kirchoff's current law has Nv−1 equations, the set for Kirchoff's voltage law has Nb−(Nv−1) equations and the set for branch voltage-current relationships has Nb equations. Thus the three sets together contain 2Nb equations.
Since for a network of Nv nodes there are only Nv−1 independent equations for Kirchoff's current law, the analyst may select which Nv−1 of the Nv nodes are used for the application of Kirchoff's current law. Likewise, the analyst may select which loops are used for the application of Kirchoff's voltage law, etc. The analyst has considerable choice regarding the nodes, loops and equations for modelling the network, as long as the analyst chooses a sufficient number of independent equations so as to permit the calculation of a unique solution. The various legitimate choices all lead to mathematically equivalent sets of network equations in which there are a total of 2Nb equations.
From the above it should be appreciated that in applications such as control system electrical network design and analysis the need arises to solve one or more systems of simultaneous linear algebraic equations (SLAE's), whose coefficient matrices may or may not be limited to merely numerical coefficients. Solutions of the SLAE's are conventionally obtained by using the well-known Gaussian elimination method. However, as described above such methods may not always work. Also, when a coefficient matrix has non-numerical coefficients, applying the Gaussian elimination method is difficult. Therefore, a need exists for improvements in this field.